pith. sign in

arxiv: 2604.07905 · v1 · submitted 2026-04-09 · 🧮 math.DG

Bertrand Legendre curves in the unit tangent bundle over Euclidean plane

Nozomi Nakatsuyama , Masatomo Takahashi This is my paper

Pith reviewed 2026-05-10 18:00 UTC · model grok-4.3

classification 🧮 math.DG
keywords Legendre curvesBertrand curvesunit tangent bundleEuclidean planeassociated curvesbijective mappingevoluteinvolute
0
0 comments X

The pith

Bertrand Legendre curves induce a bijective mapping between equivalence classes of Legendre curves in the unit tangent bundle over the Euclidean plane.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines Bertrand Legendre curves as associated curves to Legendre immersions and gives conditions under which they exist. It constructs an inverse operation that recovers the original curve from its Bertrand version. The authors then use these curves to define a natural mapping between sets of Legendre curves and prove the mapping is bijective when curves are identified under standard equivalence relations. A reader would care because the construction treats regular and singular associated curves such as evolutes and involutes inside one uniform framework.

Core claim

In the unit tangent bundle over the Euclidean plane, Bertrand Legendre curves satisfy a specific differential relation that makes them associated to any given Legendre curve. The authors supply existence conditions for both Bertrand regular plane curves and their Legendre counterparts, construct an explicit inverse operation, and prove that the induced correspondence between collections of Legendre curves is bijective modulo equivalence relations.

What carries the argument

The Bertrand Legendre curve, a Legendre immersion in the unit tangent bundle whose curvature functions obey the Bertrand linear relation, which carries the association, inversion, and bijective correspondence.

If this is right

  • Existence conditions are given for Bertrand regular plane curves and for Bertrand Legendre curves.
  • An explicit inverse operation recovers the original Legendre curve from any Bertrand Legendre curve.
  • The mapping defined by Bertrand Legendre curves is bijective on the space of Legendre curves up to equivalence relations.
  • Parallel curves, evolutes, involutes, evolutoids and involutoids arise as special cases inside the same construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The bijectivity supplies a way to generate or classify entire families of associated curves from a single representative.
  • Singularities that appear in classical evolutes or involutes remain inside the Legendre category and can be tracked uniformly.
  • Similar bijective correspondences might be constructed for Legendre curves in other ambient spaces or for higher-order associated curves.

Load-bearing premise

The curves under consideration must be Legendre immersions in the unit tangent bundle and must satisfy the stated existence conditions for the Bertrand relation.

What would settle it

An explicit example of a Legendre curve in the unit tangent bundle whose Bertrand associate cannot be inverted or for which two non-equivalent curves map to the same associate.

read the original abstract

We investigate not only the associated curves of regular plane curves, but also those of Legendre curves. As associated curves, we consider Bertrand regular plane curves and Bertrand Legendre curves. These curves contain parallel, evolute and involute curves, as well as evolutoid and involutoid curves. Since associated curves may have singular points even if the original curve is regular, Legendre curves provide a suitable framework for investigating the properties of such curves. We give existence conditions of Bertrand regular plane curves and Bertrand Legendre curves. Moreover, we give an inverse operation for Bertrand Legendre curves. Furthermore, we define a mapping between sets of Legendre curves using Bertrand Legendre curves and prove that this mapping is bijective up to equivalence relations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript studies associated curves of regular plane curves and Legendre curves in the unit tangent bundle over the Euclidean plane, with emphasis on Bertrand regular plane curves and Bertrand Legendre curves (encompassing parallel, evolute, involute, evolutoid, and involutoid curves). It states existence conditions for both classes, supplies an inverse operation for Bertrand Legendre curves, defines a mapping between sets of Legendre curves induced by the Bertrand construction, and proves that this mapping is bijective up to equivalence relations.

Significance. If the bijectivity holds under the stated immersion and regularity hypotheses, the work supplies a constructive, invertible correspondence that unifies the treatment of classical associated curves within the Legendre framework, which is particularly useful for curves with singularities. The explicit existence conditions and inverse operation add constructive value that could support further classification results or applications in singularity theory.

major comments (2)
  1. [§4] §4 (Theorem on bijectivity): the proof that the mapping is surjective relies on applying the inverse operation to an arbitrary Legendre curve, but it is not shown that the resulting curve remains a Legendre immersion when the original curve has points where the associated Bertrand curve degenerates; this assumption is load-bearing for the global bijectivity claim.
  2. [Definition 2.3 and §3] Definition 2.3 and §3: the equivalence relation used to quotient the sets of Legendre curves is defined via reparametrization and contact order, yet the verification that the Bertrand mapping descends to the quotient (i.e., is well-defined on equivalence classes) is only sketched; an explicit check that equivalent curves produce equivalent images is needed to confirm the induced map is bijective on the quotient.
minor comments (3)
  1. [§2] Notation for the unit tangent bundle and the Legendre condition (e.g., the contact form) is introduced without a self-contained reminder of the standard definitions from the literature; a brief paragraph in §2 would improve readability.
  2. [Figure 1] Figure 1 (illustrating a Bertrand Legendre curve) lacks labels for the curvature functions and the parameter t at the singular point; adding these would clarify the geometric construction.
  3. The abstract states that the mapping is bijective 'up to equivalence relations' but does not name the precise equivalence; this should be stated explicitly in the abstract for precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which highlight areas where the arguments can be made more explicit. We address each major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (Theorem on bijectivity): the proof that the mapping is surjective relies on applying the inverse operation to an arbitrary Legendre curve, but it is not shown that the resulting curve remains a Legendre immersion when the original curve has points where the associated Bertrand curve degenerates; this assumption is load-bearing for the global bijectivity claim.

    Authors: We acknowledge that the surjectivity argument in Theorem 4.1 applies the inverse construction without an explicit verification that the output remains a Legendre immersion in all cases, including potential degeneration points of the associated Bertrand curve. The definitions in the paper require the curves to satisfy the immersion and regularity conditions throughout, and the contact-order equivalence is intended to preserve these. Nevertheless, to strengthen the proof, we will revise §4 to include a direct check: given a Legendre immersion, the inverse operation (defined via the same Bertrand relation) yields a curve whose tangent and contact conditions remain non-degenerate, as the original immersion hypotheses prevent the degeneration from violating the Legendre property. revision: yes

  2. Referee: [Definition 2.3 and §3] Definition 2.3 and §3: the equivalence relation used to quotient the sets of Legendre curves is defined via reparametrization and contact order, yet the verification that the Bertrand mapping descends to the quotient (i.e., is well-defined on equivalence classes) is only sketched; an explicit check that equivalent curves produce equivalent images is needed to confirm the induced map is bijective on the quotient.

    Authors: We agree that the well-definedness of the induced map on equivalence classes is only sketched in §3. In the revised version we will supply an explicit verification: if two Legendre curves are equivalent under a reparametrization that preserves contact order, then the Bertrand images satisfy the same equivalence relation. This follows from the invariance of the Bertrand construction (and its defining differential conditions) under such reparametrizations, which we will spell out using the explicit coordinate expressions for the Legendre curves. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines Bertrand Legendre curves and related associated curves (including parallel, evolute, involute, evolutoid and involutoid), states existence conditions, supplies an explicit inverse operation, and constructs a mapping on sets of Legendre curves whose bijectivity up to equivalence relations is proved under the standing regularity and immersion hypotheses of the Legendre framework. No step reduces a claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified; the bijectivity follows directly from the supplied inverse and the equivalence relations once the geometric assumptions hold. The work is therefore a standard definitional and proof-based contribution in differential geometry with no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on standard differential-geometric axioms and introduces new curve classes by definition; no free parameters or data-fitted quantities appear.

axioms (1)
  • standard math Standard axioms of Euclidean plane geometry and the theory of Legendre curves in the unit tangent bundle
    Invoked to define the ambient space and the notion of associated curves.
invented entities (1)
  • Bertrand Legendre curve no independent evidence
    purpose: To generalize Bertrand curves so that associated curves (parallel, evolute, involute, etc.) remain well-defined even when the original curve has singularities
    Newly defined object whose properties are studied in the paper

pith-pipeline@v0.9.0 · 5411 in / 1256 out tokens · 53761 ms · 2026-05-10T18:00:40.403423+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Aminov, Differential geometry and the topology of curve s

    Y. Aminov, Differential geometry and the topology of curve s. Translated from the Russian by V. Gorkavy. Gordon and Breach Science Publishers, Amsterda m, (2000)

    work page 2000

  2. [2]

    T. M. Apostol, M. A. Mnatsakanian, Tanvolutes: generali zed involutes. Amer. Math. Monthly 117, (2010), 701–713

    work page 2010

  3. [3]

    Banchoff, S

    Y. Banchoff, S. Lovett, Differential geometry of curves and s urfaces. A K Peters, Ltd., Natick, MA, (2010)

    work page 2010

  4. [4]

    Berger, B

    M. Berger, B. Gostiaux, Differential geometry: manifolds , curves, and surfaces. Translated from the French by Silvio Levy. Graduate Texts in Mathematics, 11 5. Springer-Verlag, New York, (1988)

    work page 1988

  5. [5]

    Bertrand, M´ emoire sur la th´ eorie des courbes ` a doub le courbure

    J. Bertrand, M´ emoire sur la th´ eorie des courbes ` a doub le courbure. J. de meth´ ematiques pures et appliqu´ ees.15, (1850), 332–350

    work page

  6. [6]

    J. W. Bruce, T. J. Gaffney, Simple singularities of mapping s C,0 → C 2,0, J. London Math. Soc. (2). 26, (1982), 465–474

    work page 1982

  7. [7]

    M. P. do Carmo, Differential geometry of curves and surface s. Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N.J., (1976)

    work page 1976

  8. [8]

    Fukunaga and M

    T. Fukunaga and M. Takahashi, Existence and uniqueness f or Legendre curves. J. Geom. 104, (2013), 297–307

    work page 2013

  9. [9]

    Fukunaga, M

    T. Fukunaga, M. Takahashi, Evolutes and involutes of fro ntals in the Euclidean plane. Demonstr. Math. 48, (2015), 147–166

    work page 2015

  10. [10]

    Honda, M

    S. Honda, M. Takahashi, Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space. Turkish J. Math. 44, (2020), 883–899

    work page 2020

  11. [11]

    P. J. Giblin, J. P. Warder, Evolving evolutoids. Amer. M ath. Monthly 121, (2014), 871–889

    work page 2014

  12. [12]

    C. G. Gibson, Elementary geometry of differentiable curv es. An undergraduate introduction. Cambridge University Press, Cambridge, (2001)

    work page 2001

  13. [13]

    A. Gray, E. Abbena and S. Salamon, Modern differential geo metry of curves and surfaces with Mathematica. Third edition. Studies in Advanced Mathemati cs. Chapman and Hall/CRC, Boca Raton, FL, (2006). 17

    work page 2006

  14. [14]

    Huang, L

    J. Huang, L. Chen, S. Izumiya, D. Pei, Geometry of specia l curves and surfaces in 3-space form. J. Geom. Phys. 136, (2019), 31–38

    work page 2019

  15. [15]

    Izumiya, N

    S. Izumiya, N. Takeuchi, Generic properties of helices and Bertrand curves. J. Geom. 74, (2002), 97–109

    work page 2002

  16. [16]

    Izumiya, N

    S. Izumiya, N. Takeuchi, Evolutoids and pedaloids of pl ane curves. Note Mat. 39, (2019), 13–23

    work page 2019

  17. [17]

    K¨ uhnel, Differential geometry

    W. K¨ uhnel, Differential geometry. Curves-surfaces-ma nifolds. Translated from the 1999 German original by Bruce Hunt. Student Mathematical Library, 16. A merican Mathematical Society, Providence, RI, (2002)

    work page 1999

  18. [18]

    E. Li, D. Pei, Enveloids and involutoids of spherical Le gendre curves. J. Geom. Phys. 170, (2021), Paper No. 104371, 23 pp

    work page 2021

  19. [19]

    H. Liu, F. Wang, Mannheim partner curves in 3-space. J. G eom. 88, (2008), 120–126

    work page 2008

  20. [20]

    Nakatsuyama, Evolutes and involutes of framed curve s in the Euclidean 3-space

    N. Nakatsuyama, Evolutes and involutes of framed curve s in the Euclidean 3-space. In prepara- tion, (2026)

    work page 2026

  21. [21]

    Nakatsuyama, M

    N. Nakatsuyama, M. Takahashi, Bertrand types of regula r curves and Bertrand framed curves in the Euclidean 3-space To appear in Hokkaido Mathematical Journal, (2026)

    work page 2026

  22. [22]

    S. G. Papaioannou, D. Kiritsis, An application of Bertr and curves and surfaces to CADCAM. Computer-Aided Design. 17, (1985), 348–352

    work page 1985

  23. [23]

    I. R. Porteous, Geometric differentiation for the intell igence of curves and surfaces, Second edition. Cambridge University Press, Cambridge, (2001)

    work page 2001

  24. [24]

    D. J. Struik, Lectures on classical differential geometr y. Reprint of the second edition. Dover Publications, Inc., New York, (1988). Nozomi Nakatsuyama, Muroran Institute of Technology, Muroran 050-8585, Japan, E-mail address: 25096009b@muroran-it.ac.jp Masatomo Takahashi, Muroran Institute of Technology, Muroran 050-8585, Japan, E-mail address: masatomo...

    work page 1988